Everything everywhere all at once: a probability-based enhanced sampling approach to rare events

The problem of studying rare events is central to many areas of computer simulations. In a recent paper [Kang, P., et al., Nat. Comput. Sci. 4, 451-460, 2024], we have shown that a powerful way of solving this problem passes through the computation of the committor function, and we have demonstrated how the committor can be iteratively computed in a variational way and the transition state ensemble efficiently sampled. Here, we greatly ameliorate this procedure by combining it with a metadynamics-like enhanced sampling approach in which a logarithmic function of the committor is used as a collective variable. This integrated procedure leads to an accurate and balanced sampling of the free energy surface in which transition states and metastable basins are studied with the same thoroughness. We also show that our approach can be used in cases in which competing reactive paths are possible and intermediate metastable are encountered. In addition, we demonstrate how physical insights can be obtained from the optimized committor model and the sampled data, thus providing a full characterization of the rare event under study. We ascribe the success of this approach to the use of a probability-based description of rare events.

💡 Research Summary

The manuscript presents a novel enhanced‑sampling framework designed to tackle rare‑event problems in atomistic simulations by integrating a probability‑based description of the transition process with a metadynamics‑like bias. Building on the authors’ previous work, which introduced a variational approach to compute the committor function q(x) using a neural‑network representation and a Kolmogorov bias V_K = −(1/β) log|∇q|² to focus sampling on the transition‑state region, the current study overcomes the limitation that additional post‑processing was required to obtain the free‑energy surface (FES). The key innovation is to replace the raw committor, which is numerically ill‑behaved in the basins (q≈0 or 1) and extremely steep near the transition state, with the latent variable z(x) produced by the neural network before the sigmoid activation. The relationship q(x)=σ(z(x)) guarantees that z encodes the same information as the committor but varies smoothly across configuration space, making it a suitable collective variable (CV) for biasing.

The authors combine two biasing strategies: (i) OPES (On‑the‑fly Probability Enhanced Sampling), which uses z as a CV to flatten the underlying energy landscape and promote barrier crossing, and (ii) the static Kolmogorov bias V_K, which continuously drives the system toward configurations with large committor gradients, i.e., the transition‑state ensemble (TSE). The simultaneous application, termed OPES+V_K, yields a balanced sampling of metastable basins and the TSE, eliminates the need for separate free‑energy calculations, and provides reliable reweighting factors because the OPES bias quickly reaches a quasi‑static regime while V_K remains static.

Four test cases are examined. First, the Müller‑Brown two‑dimensional potential demonstrates that after only two iterations of the iterative scheme the method already reproduces the analytical committor, the transition‑state isoline, and the free‑energy profile with negligible error. Second, the alanine dipeptide in vacuum shows that using z‑CV dramatically improves the exploration of the φ‑ψ space compared with traditional φ/ψ CVs, captures the linear φ–θ relationship of the TSE, and yields free‑energy estimates indistinguishable from reference simulations. Third, an asymmetric double‑path potential illustrates the method’s ability to handle multiple competing pathways: the learned committor correctly reflects the lower‑barrier path’s higher flux, while the Kolmogorov distribution p_K highlights the dominant channel, avoiding the over‑representation that would arise from a naïve q≈0.5 criterion. Finally, the folding of the solvated chignolin peptide is tackled with an expanded set of 210 inter‑atomic distances as descriptors. Despite the high dimensionality, OPES+V_K converges rapidly, identifies two distinct folding routes, and reconstructs a 2‑D free‑energy surface that matches long unbiased MD references.

Beyond performance metrics, the paper emphasizes interpretability. The neural‑network parameters and the learned z‑space provide physical insight into which geometric features control the reaction progress, and the Kolmogorov bias offers a principled way to define the TSE based on the probability flux rather than an arbitrary committor value. The approach requires only the definition of the initial and final metastable states; no prior knowledge of intermediate states, reaction coordinates, or pathway multiplicity is needed.

In summary, the authors deliver a robust, semi‑automatic protocol that merges variational committor estimation with an OPES‑based metadynamics bias, using the logarithm of the committor as a collective variable. This integration yields fast convergence, balanced sampling of basins and transition states, accurate free‑energy reconstruction, and a clear physical picture of rare‑event mechanisms. The methodology is poised to become a valuable tool for studying complex biochemical reactions, material phase transitions, and catalytic processes where multiple pathways and intermediate metastable states are prevalent.